GATE 2020 IN Q.5: solution and answer

Official answer: CSource: Official answer key, GATE 2020 IN (IIT Delhi)

1. Time-domain definition. An ideal differentiator gives $y(t)=\frac{dx(t)}{dt}$.

2. Differentiation property of the Laplace/Fourier transform. $\frac{dx}{dt}\leftrightarrow sX(s)$ (with zero initial condition), so $H(s)=\frac{Y(s)}{X(s)}=s$. On the $j\omega$-axis this is $H(j\omega )=j\omega$.

3. Magnitude. $|H(j\omega )|=|j\omega |=\omega$ for $\omega \ge 0$. It is directly proportional to frequency, so it increases linearly with $\omega$: a straight line through the origin on a linear scale, or a slope of $+20\text{dB/decade}$ on a Bode plot.

4. Phase. $\angle H(j\omega )=\angle j\omega =+90^{\circ }$ for every $\omega >0$. The phase is constant, not linear in $\omega$, so (A) is false.

5. Check the other options. (B) needs $|H(j\omega )|$ independent of $\omega$ (a pure gain or an all-pass system), which $|H|=\omega$ is not. (D) needs $|H|\propto 1/\omega$, which is the integrator $H(s)=1/s$, not the differentiator.

6. Only option (C) holds.